$12^{1}_{272}$ - Minimal pinning sets
Pinning sets for 12^1_272 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_272 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 112
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7, 10, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 4, 6, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
C (optimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,6,3],[0,2,7,0],[1,7,5,1],[1,4,8,8],[2,9,7,2],[3,6,9,4],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[6,13,7,14],[19,2,20,3],[8,2,9,1],[14,5,15,6],[15,12,16,13],[3,18,4,19],[9,4,10,5],[11,16,12,17],[17,10,18,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(11,2,-12,-3)(14,7,-15,-8)(12,9,-13,-10)(3,10,-4,-11)(8,13,-9,-14)(20,15,-1,-16)(4,17,-5,-18)(18,5,-19,-6)(16,19,-17,-20)
Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-19,16)(-2,11,-4,-18,-6)(-3,-11)(-5,18)(-7,14,-9,12,2)(-8,-14)(-10,3,-12)(-13,8,-15,20,-17,4,10)(-16,-20)(1,15,7)(5,17,19)(9,13)
Loop annotated with half-edges
12^1_272 annotated with half-edges